Z2 × Z2-graded Lie Symmetries of the Lévy-Leblond Equations
N. Aizawa∗, Z. Kuznetsova†, H. Tanaka‡ and F. Toppan§
September 27, 2016
∗‡
Department of Physical Science, Graduate School of Science,
Osaka Prefecture University, Nakamozu Campus,
Sakai, Osaka 599-8531 Japan.
† UFABC, Av. dos Estados 5001, Bangu,
cep 09210-580, Santo André (SP), Brazil.
§ CBPF, Rua Dr. Xavier Sigaud 150, Urca,
cep 22290-180, Rio de Janeiro (RJ), Brazil.
Abstract
The first-order differential Lévy-Leblond equations (LLE’s) are the non-relativistic analogs
of the Dirac equation, being square roots of (1 + d)-dimensional Schrödinger or heat equations. Just like the Dirac equation, the LLE’s possess a natural supersymmetry. In previous
works it was shown that non supersymmetric PDE’s (notably, the Schrödinger equations for
free particles or in the presence of a harmonic potential), admit a natural Z2 -graded Lie
symmetry.
In this paper we show that, for a certain class of supersymmetric PDE’s, a natural Z2 ×Z2 graded Lie symmetry appears. In particular, we exhaustively investigate the symmetries of
the (1 + 1)-dimensional Lévy-Leblond Equations, both in the free case and for the harmonic
potential. In the free case a Z2 × Z2 -graded Lie superalgebra, realized by first and secondorder differential symmetry operators, is found. In the presence of a non-vanishing quadratic
potential, the Schrödinger invariance is maintained, while the Z2 - and Z2 × Z2 - graded
extensions are no longer allowed.
The construction of the Z2 × Z2 -graded Lie symmetry of the (1 + 2)-dimensional free heat
LLE introduces a new feature, explaining the existence of first-order differential symmetry
operators not entering the super Schrödinger algebra.
CBPF-NF-004/16
∗
E-mail:
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CBPF-NF-004/16
1
1
Introduction
In this paper we prove the existence of a finite Z2 × Z2 -graded Lie symmetry, realized by firstorder and second-order differential operators, of the free generalized Lévy-Leblond equations
(the (1 + 1)- and (1 + 2)-dimensional cases are here explicitly discussed).
It was pointed out in [1] that finite Lie superalgebras can be symmetries for a certain class
of purely bosonic partial differential equations (including the cases of the free particle and of
the harmonic oscillator)∗ . The recognition that a symmetry superalgebra is present was later
applied [3] in the context of Conformal Galilei Algebras to identify new bosonic invariant partial
differential equations.
It is therefore natural to pose the question: what happens in the case of a supersymmetric
system, namely one which already possesses a Z2 -graded structure? Is, in that case, a second
Z2 -gradation present? To give an answer we started investigating the generalized Lévy-Leblond
equations, which are the non-relativistic analogs of the Dirac equations, being the square roots
of the heat or of the Schrödinger equation (with or without potential terms). The original LévyLeblond equation [4] is the non-relativistic wave equation of a spin- 12 particle in the ordinary
(1 + 3)-dimensional space-time and possesses a natural supersymmetry. In Section 2 we discuss
at length the generalized Lévy-Leblond equations, induced by first-order matrix differential
operators, and their construction from the relevant Clifford algebras.
This investigation about graded symmetries requires a preliminary understanding of the
Z2 × Z2 -graded Lie (super)algebra structures. Generalizations of Lie and super-Lie algebras
were introduced and named color algebras (for certain resemblances to parastatistics) in [5, 6].
They were further investigated in [7, 8]. Nowadays there is a small body of literature about
these structures, dealing with possible physical applications (see, e.g., [9, 10, 11, 12]) and a
larger number of works devoted to their mathematical development (for more recent papers see,
e.g., [13, 14] and references therein). The absence of a spin-statistics connection in the relativistic
setting (since we are working in a non-relativistic setting, this is not a concern for us) has been
the major impediment for the full development of the topic of generalized supersymmetries. To
make this paper self-contained, the relevant properties of Z2 ×Z2 -graded color Lie (super)algebras
are presented in an Appendix.
Our investigation requires the further notion, following [15], of symmetry operator. We
consider two classes of symmetries operators; they can be recovered either from commutators or
from anticommutators, see equations (14) and, respectively, (15). Quite interestingly, we need
symmetry operators belonging to both classes in order to produce a Z2 × Z2 -graded symmetry.
In this work we computed the complete list of symmetry operators for the Lévy-Leblond
square root of the free heat equation in (1 + 1)-dimensions (a 2 × 2 matrix operator) and the
square root of the free Schrödinger equation in (1 + 1)-dimensions (a 4 × 4 matrix operator in
the real counting). We also computed the symmetry operators recovered from commutators for
the Lévy-Leblond square root of the heat equation with quadratic potential (a 4 × 4 matrix
operator). Once identified the symmetry operators, we investigated the closed, finite (graded)
Lie symmetry algebras induced by them. We proved, in particular, that the Lévy-Leblond
square roots of the (1 + 1)-dimensional free heat and free Schrödinger equations possess a super
Schrödinger symmetry algebra with maximal N = 1 extension (super Schrödinger algebras
were discussed in [16, 17]). The Z2 × Z2 -graded Lie symmetry GZ2 ×Z2 of first and second-order
∗
The fact that a Lie superalgebra appears even in a purely bosonic setting is not so surprising. Indeed,
for the harmonic oscillator, the old results of [2] can be expressed, in modern language, by stating that the
Fock vacuum of creation/annihilation operators can be replaced by a lowest weight representation of an osp(1|2)
spectrum-generating superalgebra.
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differential operators, spanned by the 13 generators in (53), is found.
The situation is quite different for the square root of the heat equation with quadratic
potential. A Schrödinger symmetry algebra is still present. We proved that it can not, on the
other hand, be extended to a super Schrödinger symmetry algebra.
In Appendix B we present the construction of the Z2 ×Z2 -graded Lie symmetry superalgebra
of the Lévy-Leblond operator associated with the (1 + 2)-dimensional free heat equation. A
new feature emerges. The Z2 × Z2 -graded symmetry includes first-order differential symmetry
operators which do not belong to the two-dimensional super Schrödinger algebra.
We postpone to the Conclusions a more detailed summary of our results, with comments
and a discussion of their implications.
The scheme of the paper is as follows. In Section 2 we introduce the generalized LévyLeblond operators and their relation to Clifford algebras. In Section 3 we introduce, following
[15], the notion of symmetry operators. In Section 4 the full list of symmetry operators of the
Lévy-Leblond square roots of the free heat and free Schrödinger equation in (1 + 1)-dimensions
is presented. Finite (graded) Lie algebras induced by these symmetry operators (including the
N = 1 super Schrödinger algebra and the Z2 × Z2 -graded symmetry) are presented in Section 5.
Section 6 is devoted to the symmetry of the Lévy-Leblond square root of the (1 + 1)-dimensional
heat equation with quartic potential. A summary of our results is given in the Conclusions. A
self-contained presentation of the Z2 × Z2 -graded color Lie (super)algebras (and their relation
to quaternions and split-quaternions) is given in Appendix A. We present in Appendix B the
construction of the Z2 × Z2 -graded symmetry of the free (1 + 2)-dimensional Lévy-Leblond
equation. In Appendix C we discuss the possibility of introducing different Z2 × Z2 -graded Lie
(super)algebras, based on different assignments of gradings to given operators.
2
On Clifford algebras and generalized Lévy-Leblond equations
The original Lévy-Leblond equation [4] is the square root of the Schrödinger equation in 1 +
3 dimensions. Generalized Lévy-Leblond equations are square roots of heat or Schrödinger
equations in an arbitrary number of space dimensions; they can be systematically constructed
from Clifford algebras irreducible representations. We introduce here the basic ingredients for
this general scheme, focusing on issues such as complex structure, introduction of a potential
term and so on.
We remind that the irreducible representations (over R) of the Cl(p, q) Clifford algebras (the
enveloping algebras whose gamma-matrix generators satisfy γi γj + γj γi = 2ηij , where ηij is a
diagonal matrix with p positive (+1) and q negative (−1) entries) can be obtained by tensoring
four 2 × 2 matrices, see e.g. [18]. It is convenient to follow the presentation of [19]. The 2 × 2
matrices can be identified with (four) letters. General gamma matrices can be expressed, since
no ambiguity arises, as words in a 4-letter alphabet by dropping the symbol of tensor product
“⊗”. By expressing
1 0
1 0
0 1
0 1
I=
, X=
, Y =
, A=
,
(1)
0 1
0 −1
1 0
−1 0
the split-quaternions, see Appendix A, can be represented as
ee0 = I,
ee1 = Y,
ee2 = X,
ee3 = A,
where X, Y, A are the gamma matrices realizing the Cl(2, 1) Clifford algebra.
(2)
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The quaternions can be realized as 4 × 4 real matrices. With the adopted convention of
dropping the tensor product symbol they can be presented as
e0 = II,
e1 = AI,
e2 = XA,
e3 = Y A.
(3)
Some comments are in order:
i) a matrix is block-diagonal if, in its associated word, the first letter is either I or X. Conversely,
it is block-antidiagonal if the first letter is Y or A,
ii) a matrix is (anti)symmetric, depending on the number (even or odd) of A’s in its word,
iii) in real form the complex structure is defined by a real matrix J such that J 2 = −I.
The complex-structure preserving matrices commute with J. If the complex structure is preserved, complex numbers can be used. A possible choice for the 4 × 4 real matrices complex
structure is setting J = IA, since (IA)2 = −II = −I4 . Therefore, the eight 4 × 4 complexstructure preserving matrices are II, XI, Y I, AA, IA, XA, Y A, AI.
Depending on the (p, q) signature, the maximal number p + q of Clifford gamma-matrix
generators that can be accommodated in 2n × 2n real matrices is
2 × 2 : Cl(2, 1),
4 × 4 : Cl(3, 2),
Cl(0, 3),
8 × 8 : Cl(4, 3),
Cl(5, 0),
Cl(1, 4),
Cl(0, 7),
(4)
and so on.
Clifford algebras can be used to introduce Supersymmetric Quantum Mechanics (SQM). In
its simplest version (the one-dimensional, N = 2 supersymmetry, with x as a space coordinate)
the two supersymmetry operators Q1 , Q2 need to be block anti-diagonal, hermitian and complex
structure preserving first-order real differential operators. Moreover, they have to satisfy the
N = 2 SQM algebra
{Qi , Qj } = 2δij H,
[H, Qi ] = 0,
(5)
where H is the Hamiltonian.
The irreducible representation (in real counting) requires 4 × 4 matrices. By setting the
complex structure to be J = IA, the most general solution, for an arbitrary function f (x) (the
prepotential), can be expressed as
Q1 = AI∂x + Y If (x),
Q2 = Y A∂x + AAf (x),
H = II(−∂x2 + f (x)2 ) + XIfx (x) = I4 (−∂x2 + f (x)2 ) + Nf fx (x).
(6)
The hamiltonian H is diagonal. In its upper block the potential is V+ (x) = f (x)2 + fx (x), while
in its lower block the potential is V− (x) = f (x)2 − fx (x).
XI defines the fermion number operator Nf (Nf = XI). All operators act on two real
component bosonic and two real component fermionic fields. For energy eigenvalues E > 0 the
H± = −∂x2 + f (x)2 ± fx (x) Hamiltonians share the same spectrum. The free case is obtained by
taking f (x) = 0; the harmonic oscillator case is recovered by taking f (x) = cx.
Since Q1 , Q2 , H commute with J and preserve the complex structure, they can also be
expressed as 2 × 2 complex matrices.
The construction of a (generalized) Lévy-Leblond real differential operator requires the linear
combination of two gamma matrices (one with a positive and the other one with a negative
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square) in order to produce a nilpotent operator which eliminates the ∂t2 term present in the
relativistic Klein-Gordon equation.
The original Lévy-Leblond equation (the square root of the Schrödinger equation in 1 + 3
dimensions), can be recovered from the Cl(1, 4) Clifford algebra, whose matrices act on 8 real
component fields. Since Cl(1, 4) has a quaternionic (and therefore, a fortiori, complex) structure
[18], the 8 × 8 real matrices preserving the complex structure can be described as 4 × 4 complex
matrices acting on a multiplet of 4 component complex fields.
We call a first-order real matrix differential operator a “generalized Lévy-Leblond operator” if
it is the square root of either the heat or the Schrödinger equation in 1 + d dimensions (with
or without a potential term). Further properties can be imposed. It could be required the
operator to be block anti-diagonal and anticommuting with the fermion number operator, so
that it mutually exchanges bosons into fermions.
The minimal matrix size to accommodate a Lévy-Leblond square root of the 1 + 1 heat
equation is 2. Indeed, we can set
ΩΨ(x, t) = 0,
1
1
2
Ω = (A + Y )∂t + (A − Y )λ + X∂x → Ω = I2 (−λ∂t + ∂x2 ),
2
2
(7)
with λ an arbitrary real number.
One should note that Ω is neither block-antidiagonal nor complex structure preserving.
The minimal solution to have a block antidiagonal (and complex-structure preserving) operator
requires 4 × 4 real matrices. A convenient basis for the five 4 × 4 Cl(3, 2) gamma matrices is
given by AA, AX, AY, XI, Y I. The complex structure can be defined by J = AI. The three
complex-structure preserving matrices AA, AX, AY satisfy the p = 1, q = 2 gamma-matrix
relations.
A block antidiagonal, complex structure preserving, square root of the free heat equation in
1 + 1 dimensions is given by
1
1
(AA + AY )∂t + (AA − AY )λ + AX∂x ,
2
2
= II(λ∂t − ∂x2 ) = I4 (λ∂t − ∂x2 ).
Ωheat,f ree =
Ω2heat,f ree
(8)
The introduction of a potential term requires the use of the extra antidiagonal gamma matrix
Y I. Therefore, in the presence of a non-vanishing potential, the 4 × 4 Lévy-Leblond operator
cannot preserve the complex-structure. We have
1
1
(AA + AY )∂t + (AA − AY )λ + AX∂x + Y If (x),
2
2
2
= II(λ∂t − ∂x + f (x)2 ) + XXfx (x).
Ωheat =
Ω2heat
(9)
Due to the mentioned property, Ωheatf ree in (8) can be represented by 2 × 2 complex matrices,
while this is not true for Ωheat in (9) if f (x) is non-vanishing.
The (free) Schrödinger equation requires the complex structure. The square root of the free
Schrödinger equation can be obtained by modifying Ωheat,f ree so that
ΩSch,f ree =
1
1
(AA + AY ) · AI∂t + (AA − AY )λ + AX∂x ,
2
2
(10)
where “·” denotes the ordinary matrix multiplication.
Its square gives
Ω2Sch,f ree = AIλ∂t − II∂x2 ,
(11)
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namely a Schrödinger equation on 4-component real fields (which can be equivalently expressed
as an equation for 2-component complex fields).
The introduction of a non-vanishing potential for the 1 + 1 Schrödinger equation requires
a Lévy-Leblond operator ΩSch acting on at least 8 real component fields. With the adopted
conventions ΩSch can be expressed as
1
1
(AAA + AAY ) · AII∂t + (AAA − AAY )λ + AY I∂x + AAXf (x),
2
2
2
= −AIIλ∂t + III(−∂x + f (x)2 ) + IXXfx (x),
ΩSch =
Ω2Sch
(12)
the complex structure being defined by J = AII.
It is straightforward to systematically construct, following this scheme and with the tools in
[19], generalized Lévy-Leblond operators in (1 + d) dimensions.
3
On symmetries of matrix partial differential equations
Our investigation heavily relies on the notion of symmetry operator defined in [15]. We recall
that a symmetry operator Z of a matrix partial differential equation induced by an operator Ω
maps a given solution Ψ(~x) into another solution ZΨ(~x), according to
ΩΨ(~x) = 0 ⇒ Ω(ZΨ(~x))|ΩΨ=0 = 0.
(13)
Z can be any kind of operator. In the traditional Lie viewpoint the symmetry group of a
differential equation is generated by a subset of operators which are closed under commutation.
This condition is relaxed for superalgebras or Z2 × Z2 graded Lie algebras. Based on the grading
of the symmetry operators, the closure requires both commutators and anticommutators.
If we restrict Z to be a differential operator of finite order, Z can be called a symmetry
operator [15] if the following sufficient condition for symmetry is fulfilled:
either
[Ω, Z] = ΦZ (~x)Ω,
(14)
or
{Ω, Z} = ΦZ (~x)Ω,
(15)
where ΦZ (~x) is a n × n matrix-valued function of the ~x space(time) coordinates.
If there is no ambiguity (certain operators, like the identity I, satisfy both (14) and (15))
throughout the text we denote with Σ’s the symmetry operators satisfying (14) and with Λ’s
the symmetry operators satisfying (15).
In our applications we consider, initially, first-order differential symmetry operators. Secondorder differentials symmetry operators are also constructed by taking suitable anticommutators
of the previous operators.
It is important to note that, if Σ1 , Σ2 are two operators satisfying (14), then by construction
the identity
[Ω, [Σ1 , Σ2 ]] = Φ[Σ1 ,Σ2 ] Ω
(16)
is satisfied. This does not imply, however, that the commutator [Σ1 , Σ2 ] is a symmetry operator
according to the given definition. Indeed, Φ[Σ1 ,Σ2 ] can be a differential operator and not necessarily a matrix-valued function. In the following, see e.g. (38,39), some examples are given. A
Lie algebra of symmetry operators requires Φ[Σ1 ,Σ2 ] to be a matrix-valued function.
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4
Symmetries of the Lévy-Leblond square root of the free heat
and Schrödinger equations
As discussed in Section 2, the minimal realization of the Lévy-Leblond square root of the 1 + 1dimensional free heat (Schrödinger) equation requires 2 × 2 (and, respectively, 4 × 4) matrices.
We produce here the exhaustive list of its symmetry operators.
With respect to Section 2 we slightly change the notations in order to easily extrapolate the
results from the heat to the Schrödinger equation. It is convenient to set, as 2 × 2 matrices,
1
γ± = (Y ± A) , γ3 = X,
2
(17)
where X, Y, A and the identity I ≡ I are introduced in (1). The following relations are satisfied
2 = 0,
γ±
γ32 = I,
γ± γ∓ = 12 (I ± γ3 ),
γ3 γ± = ±γ± = −γ± γ3 .
(18)
Our analysis of the symmetries only uses the (18) relations; therefore, the results below are
representation-independent and can be extended to other representations (in particular to the
case of 4 × 4 matrices). For 4 × 4 matrices we can introduce, following the convention of Section
2, the Cl(1, 2) Clifford algebra generators
γ
e1 = AA,
γ
e2 = AY,
γ
e3 = AX.
(19)
They define the complex structure J given by
J
= γ
e1 γ
e2 γ
e3 = −AI,
(J 2 = −II = −I).
(20)
A realization of the (18) algebra in terms of 4 × 4 matrices is given by setting
γ± = ± 21 J · (e
γ1 ± γ
e2 ),
γ3 = J · γ
e3 .
(21)
By construction γ± , γ3 defined in (21) commute with J ([J, γ± ] = [J, γ3 ] = 0).
The Lévy-Leblond operator Ω of the free heat equation is introduced (both in the 2 × 2 and
in the 4 × 4 representations) as
Ω = γ+ ∂t + γ− λ + γ3 ∂x .
(22)
Ω2 = I · (λ∂t + ∂x2 ).
(23)
Its square is
According to the convention introduced in Section 3 we denote with Σ’s the symmetry operators
arising from commutators and with Λ’s the symmetry operators arising from anticommutators
[Σ, Ω] = ΦΣ · Ω,
(24)
{Λ, Ω} = ΦΛ · Ω,
(25)
for some given matrix-valued function ΦΣ or ΦΛ in the x, t variables.
The computation of the symmetry operators is tedious but straightforward. In the 2 × 2
case the complete list (up to normalization) of first-order differential operators satisfying (24)
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is given by
H = I∂t ,
1
1
1
D = −(I(t∂t + x∂x + ) + γ3 ),
2
2
4
λ
1
1
K = −(I(t2 ∂t + tx∂x − x2 + t) − γ+ x + γ3 t),
4
2
2
P+ = I∂x ,
1
λ
P− = I(t∂x − x) − γ+ ,
2
2
C = I,
Ωz(x,t) = z(x, t)Ω = z(x, t)(γ+ ∂t + γ− λ + γ3 ∂x ).
(26)
The class of symmetry operators Ωz(x,t) depends on an unconstrained function z(x, t) (Ω is
recovered by setting z(x, t) = 1).
The complete set (up to normalization) of 2 × 2 first-order differential operators satisfying
(25) is given by
Λ1 = I,
Λ2 = γ+ ∂t − λγ− ,
1
λ
Λ3 = γ+ (t∂t + ) − γ− λt + xI,
2
2
Λ4 = γ3 ∂t − 2γ− ∂x ,
λ
x 1
1
Λ5 = γ3 (t∂t + ) + γ− (−2t∂x + ) − γ+ + I,
4
2
2 2
1 2
t
λ
1
x
t
λx2
Λ6 = γ3 ( t ∂t + ) + γ− (−t2 ∂x + tx) + γ+ (− tx∂t − ) + ( −
)I,
2
4
2
2
4
2
8
e w(x,t) = w(x, t)(γ+ ∂x − λ (I + γ3 )).
Λ
2
(27)
e w(x,t) class of operators depend on an unconstrained function w(x, t).
The Λ
The computation of the associated matrix-valued functions ΦΣ (x, t), ΦΛ (x, t) is left as a
simple exercise for the Reader.
As explained above, formulas (26) and (27) are representation-independent and provide
symmetry operators for the 4 × 4 matrix case as well. For this matrix representation, since
the complex structure operator J commutes with γ± , γ3 , the number of symmetry operators
is doubled with respect to the 2 × 2 matrix representation. Indeed, for any given symmetry
operator Σ or Λ entering (26) or (27), J · Σ (J · Λ) is a symmetry operator satisfying (24) (and,
respectively, (25)).
We explicitly verified that no further symmetry operator is encountered, in the 4 × 4 representation, besides the operators given in (26,27) and their J-doubled counterparts.
The presence of the complex structure J in the 4 × 4 matrix case allows to induce, from the
Lévy-Leblond square root of the free heat equation, the Lévy-Leblond square root of the free
Schrödinger equation in 1 + 1 dimensions. The most straightforward way consists in replacing,
in the formulas above, λ 7→ βJ, for a real β. This substitution is allowed because J commutes
with all matrices entering (22,26,27).
We therefore get, for the 1 + 1-dimensional free Schrödinger case,
Ω = γ+ ∂t + γ− · Jβ + γ3 ∂x ,
Ω
2
= I · (Jβ∂t + ∂x2 ).
(28)
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5
The graded Lie symmetry algebras of the free equations
We investigate here the properties of the closed graded Lie (super)algebras recovered from the
symmetry operators entering (26) and (27).
We start by observing that the first six operators in (26) span the one-dimensional Schrödinger
algebra sch(1). Their non-vanishing commutators are
[D, H] = H,
[D, K] = −K,
[H, K] = 2D,
1
[D, P± ] = ± P± ,
2
[H, P− ] = P+ ,
[K, P+ ] = P− ,
λ
[P+ , P− ] = − C.
2
(29)
D is the grading operator, the grading z being defined by the commutator
[D, Z] = zZ,
(30)
for some given generator Z.
H, D, K close an sl(2) algebra with D as the Cartan element.
P± , together with the central charge C, realize the h(1) Lie-Heisenberg subalgebra. Since
P± have half-integer grading, it is convenient in the following to introduce the notation
P± ≡ P± 1 .
(31)
2
P± 1 induce a closed osp(1|2) algebra, whose even generators are the second-order differential
2
operators P±1 , P0 , introduced via the anticommutators (for δ and taking values ±1)
P(δ+) 1
2
= {Pδ 1 , P 1 },
2
(32)
2
while P± 1 belong to the odd sector of osp(1|2).
2
One should note that [D, Ps ] = sPs . Since
[Ω, Ps ] = 0,
1
∀s = ±1, ± , 0,
2
(33)
the osp(1|2) Lie superalgebra spanned by the Ps ’s operators is a symmetry superalgebra of the
Lévy-Leblond operator Ω.
A closed symmetry superalgebra u(1) ⊕ sl(2)⊃
+osp(1|2), with 2 odd and 7 even generators,
is spanned by H, D, K, C and the five Ps operators.
We note, as a remark, that if we express the free heat equation
Ω2 Ψ(x, t) = (λ∂t + ∂x2 )Ψ(x, t) = 0
(34)
in the equivalent form
−λ∂t Ψ(x, t) = ∂x2 Ψ(x, t),
(35)
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the operator P 1 is the square root of the r.h.s. operator in (35).
2
The Lévy-Leblond operator Ω ≡ Ωz(x,t)=1 possesses half-integer grading:
[D, Ω] =
1
Ω.
2
(36)
The commutator with K produces an operator with − 12 grading,
[K, Ω] = tΩ ≡ Ωz(x,t)=t ,
1
[D, tΩ] = − tΩ.
2
(37)
It is convenient to set Ω ≡ Ω 1 , tΩ ≡ Ω− 1 .
2
2
The commutator
[Ω, tΩ] = γ+ Ω
(38)
gives an operator, γ+ Ω, which does not belong to the class of symmetry operators here considered. The reason is that its commutator with Ω can be expressed as
[Ω, γ+ Ω] = (−λγ3 + 2γ+ ∂x )Ω ≡ T Ω,
(39)
where T = −λγ3 + 2γ+ ∂x is not a matrix-valued function, but a differential operator.
On the other hand, Ω± 1 can be taken as the odd sector of an osp(1|2) superalgebra whose
2
even generators are the second-order differential operators Ω±1 , Ω0 given by the anticommutators
(for δ and taking values ±1)
Ω(δ+) 1
2
= {Ωδ 1 , Ω 1 }.
2
(40)
2
The non-vanishing commutators of this second osp(1|2) superalgebra are given by
[Ω0 , Ωs ] = −2sλΩs ,
1
s = ± , ±1, 0,
2
[Ω1 , Ω−1 ] = 4λΩ0 ,
[Ω± 1 , Ω∓1 ] = ±2λΩ∓ 1 .
2
(41)
2
It follows, as a corollary, that the second-order differential operators Ω±1 , Ω0 are symmetry
operators (since, e.g., [Ω−1 , Ω 1 ] = −2λtΩ 1 ).
2
2
It is worth pointing out that in the Schrödinger case, when λ is replaced by βJ, see the
discussion before equation (28), the five symmetry operators closing the osp(1|2) superalgebra
are Ω+ 1 , J · Ω− 1 , Ω±1 , J · Ω0 .
2
2
The action of the sch(1) Schrödinger generators H, D, K, C, P± 1 on the five operators Ωs
2
produces the semidirect superalgebra sch(1)⊃
+osp(1|2) since the only non-vanishing commutators
involving the Ωs operators and the Schrödinger generators are
1
s = ± , ±1, 0,
2
[H, Ω−1 ] = 2Ω0 ,
[D, Ωs ] = sΩs ,
[H, Ω− 1 ] = Ω 1 ,
2
2
[K, Ω 1 ] = −Ω− 1 ,
2
2
[K, Ω0 ] = Ω−1 ,
[H, Ω0 ] = Ω1 ,
[K, Ω1 ] = 2Ω0 .
(42)
10
CBPF-NF-004/16
Due to the fact that, for any s, s0 = ± 21 , ±1, 0,
[Ps , Ωs0 ] = 0,
(43)
a first Z2 × Z2 -graded Lie superalgebra, whose vector space is spanned by the Ps and Ωs0
symmetry operators, is obtained. This Z2 × Z2 -graded Lie superalgebra is decomposed (see
Appendix A, equations (A.1,A.2,A.4)) according to
G00 = {P±1 , P0 , Ω±1 , Ω0 },
G01 = {P± 1 },
2
G10 = {Ω± 1 },
G11 = {∅},
2
(44)
with an empty G11 sector.
A super Schrödinger symmetry algebra is obtained. In order to recover it, we need to
introduce the symmetry operators arising from the Λ-sector defined by the equation (25). The
presence of a super Schrödinger algebra requires the existence of symmetry operators which are
square roots, up to a sign, of H, K entering (26,29). In the 4 × 4 representation, due to the
existence of the complex structure J, the sign can be flipped (if Q is an operator such that
Q2 = ±H, then J · Q is a symmetry operator such that (J · Q)2 = ∓H). Our analysis proves
the existence of a super Schrödinger algebra with at most an N = 1 supersymmetric extension,
the only supersymmetric roots being the symmetry operators Q± given by
1
1
√ Λ2 = √ (γ+ ∂t − λγ− ),
λ
λ
Q+ =
Q2+ = −H,
(45)
and
Q− =
1
e x ) = √1 (γ+ (t∂t + x∂x + 1 ) − γ− λt − γ3 λx ),
√ (Λ3 + Λ
2
2
λ
λ
Q2− = K.
(46)
Since Q± have half-integer grading,
1
[D, Q± ] = ± Q± ,
2
(47)
as before it is convenient to set Q± ≡ Q± 1 .
2
The operators Q± 1 induce a third osp(1|2) symmetry superalgebra. Their anticommutators
2
Q(δ+) 1 = {Qδ 1 , Q 1 }, for δ, = ±1, produce the even generator Q±1 , Q0 which coincides, up
2
2
2
to normalization, with the sl(2) generators H, K, D given in (26):
{Q+ 1 , Q+ 1 } = −2H,
2
2
{Q− 1 , Q− 1 } = 2K,
2
2
{Q+ 1 , Q− 1 } = 2D.
2
(48)
2
The remaining non-vanishing commutators of the third osp(1|2) superalgebra are
[H, Q− 1 ] = Q+ 1
2
2
, [K, Q+ 1 ] = Q− 1 .
2
(49)
2
In the super Schrödinger algebra, the two extra operators P± 1 enter the even sector and a
2
further symmetry operator X, obtained from the commutators
[P± 1 , Q∓ 1 ] = ±X,
2
(50)
2
enters the odd sector. We have
X =
√
1 e
λ
1
λ
√ Λw(x,t)=1 +
Λ1 = √ (γ+ ∂x − γ3 ).
2
2
λ
λ
(51)
11
CBPF-NF-004/16
The super Schrödinger algebra ssch(1) is decomposed into even and odd sector as
ssch(1) = G0 ⊕ G1 ,
with G0 = {H, D, K, C, P± 1 } and G1 = {Q± 1 , X}.
2
2
The extra non-vanishing (anti)commutators involving the X generator are
{X, Q± 1 } = −P± 1 ,
2
2
{X, X} =
λ
C.
2
(52)
It is worth pointing out that the operator Q+ 1 is the square root of the operator in the l.h.s. of
2
the equation (35).
We recovered three independent osp(1|2) symmetry superalgebras induced, respectively, by
the three pairs of half-graded odd generators P± 1 , Ω± 1 , Q± 1 . We already discussed the com2
2
2
patibility of the osp(1|2) superalgebras obtained from the two pairs P± 1 , Ω± 1 . The last step
2
2
consists in discussing the compatibility of the osp(1|2) superalgebras induced by the two pairs
Q± 1 , Ω± 1 and Q± 1 , P± 1 .
2
2
2
2
It is easily shown that an algebra which presents both pairs of Q± 1 , Ω± 1 generators brings us
2
2
beyond the scheme of symmetry algebras discussed in the present paper. Indeed, their repeated
(anti)commutators produce higher derivative operators which do not satisfy equations (24) and
(25) with matrix-valued functions on their right hand sides.
On the other hand, requiring the presence of the two pairs of operators Q± 1 , P± 1 , a Z2 × Z2 2
2
graded Lie superalgebra (following the definition in Appendix A) of symmetry operators is
naturally induced. This superalgebra, GZ2 ×Z2 , is decomposed according to GZ2 ×Z2 = G00 ⊕ G01 ⊕
G10 ⊕ G11 , with their respective sectors spanned by the generators
G00 = {H, D, K, P±1 , P0 },
G01 = {P± 1 },
2
G10 = {Q± 1 , X± 1 },
2
2
G11 = {X}.
(53)
The extra generators we have yet to introduce are the second-order differential operators X± 1 ,
2
obtained from the anticommutators
X± 1
2
= {X, P± 1 }.
(54)
2
Some comments are in order. The Z2 × Z2 -graded Lie superalgebra (53) is closed under (anti)commutators and, furthermore, the Z2 × Z2 -graded Jacobi identities are fulfilled, as explicitly
verified.
It is important to point out that (53) is a Z2 × Z2 -graded Lie superalgebra of symmetry
operators (as defined in Section 3). The last check consists in verifying that the anticommutators
of X± 1 with Ω ≡ Ω+ 1 are vanishing:
2
2
{Ω, X} = {Ω, X± 1 } = 0.
(55)
2
For completeness we present the list of the remaining (besides the ones already introduced)
non-vanishing (anti)commutators involving the (53) generators. We have
12
CBPF-NF-004/16
[D, Ps ] = sPs ,
[D, X± 1 ] = ± 21 X± 1 ,
[H, Pm ] = (1 − m)Pm+1 ,
[H, X− 1 ] = X 1 ,
[K, Pm ] = (1 + m)Pm−1 ,
[K, X 1 ] = X− 1 ,
[P1 , P−1 ] = −4λP0 ,
[P±1 , P∓ 1 ] = ∓2λP± 1 ,
[P±1 , Q∓ 1 ] = ±2X± 1 ,
[P±1 , X∓ 1 ] = ∓2λX± 1 ,
[P0 , Ps ] = 2sλPs ,
[P0 , Q± 1 ] = ∓X± 1 ,
[P0 , X± 1 ] = ±λX± 1 ,
[P± 1 , Q∓ 1 ] = ±X,
[P± 1 , X∓ 1 ] = ∓λX,
{X± 1 , X± 1 } = λP±1 ,
{X 1 , X− 1 } = λP0 ,
{Q± 1 , X± 1 } = −P±1 ,
{Q± 1 , X∓ 1 } = −P0 ,
{X, X± 1 } = λP± 1 .
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
(56)
2
2
2
2
2
2
2
2
where m = ±1, 0 and s = ± 21 , ±1, 0.
It is worth pointing out that adding the identity operator C to the G00 sector, we obtain the
Z2 × Z2 -graded Lie superalgebra u(1) ⊕ GZ2 ×Z2 . The N = 1 super Schrödinger algebra ssch(1)
is spanned by a subset of the u(1) ⊕ GZ2 ×Z2 generators. Even so, ssch(1) is not a u(1) ⊕ GZ2 ×Z2
subalgebra (since for ssch(1) the brackets involving, e.g., P± 1 are defined with commutators).
2
6
Symmetries of the Lévy-Leblond square root of the heat equation with quadratic potential
The Schrödinger algebra is the maximal kinetic symmetry algebra of the heat or of the Schrödinger
equation, see [20, 21, 1], for three type of potentials, constant, linear or quadratic. The last case
corresponds to the harmonic oscillator. We investigate here the symmetries of the the associated Lévy-Leblond equation for the heat equation with quadratic potential. On the basis of
the construction of Section 2, the minimal matrix differential operator is expressed as a 4 × 4
matrix. The quadratic potential is recovered from the linear function f (x) = ωx entering (9).
Without loss of generality we can set λ = ω = 1 and take the block-antidiagonal Lévy-Leblond
e to be given by
operator Ω
e = (e14 − e32 )∂t + (e13 − e24 − e31 + e42 )∂x + x(e13 + e24 + e31 + e42 ) − e23 + e41 , (57)
Ω
where eij denotes the matrix with entry 1 at the cross of the i-th row and j-th column and 0
otherwise.
e 2 is
In this basis its squared operator Ω
e 2 = (e11 + e22 + e33 + e44 )(∂t − ∂ 2 + x2 ) + e11 + e44 − e22 − e33 .
Ω
x
(58)
The exhaustive list of first-order differential symmetry operators Σ’s satisfying
e = ΦΣ (x, t) · Ω,
e
[Σ, Ω]
(59)
for some 4 × 4 matrix-valued functions ΦΣ (x, t) in the x, t coordinates, can be computed with
lengthy but straightforward methods. We present here the complete list of symmetry operators
(for simplicity we leave as an exercise for the Reader the computation of their associated matrixvalued functions ΦΣ (x, t)’s).
13
CBPF-NF-004/16
The symmetry operators can be split into the two big classes of block-diagonal and blockantidiagonal operators. In both cases we have 12 fixed symmetry operators (up to normalization)
plus two extra sets of operators depending on an unconstrained function (denoted as z(x, t)) of
the x, t coordinates.
The 12 fixed block-diagonal symmetry operators can be conveniently presented as
Σ1 = e4t ((e22 + e33 − 2xe34 )∂t + (e21 − e43 + 2x(e22 + e44 ))∂x +
xe21 + 2x2 e22 + 4e33 − 8xe34 − xe43 + 2(x2 + 1)e44 ),
Σ2 = e4t ((e11 + e44 + 2xe34 )∂t + (e43 − e21 + 2x(e11 + e33 ))∂x +
(2 + 2x2 )e11 − xe21 + 2x2 e33 + xe43 ),
Σ3 = e−4t ((e22 + e33 + 2xe34 )∂t + (e21 − e43 − 2x(e22 + e44 ))∂x +
−3xe21 − 2x2 e22 + (4x2 − 2)e33 − xe43 + 2x2 e44 ),
Σ4 = e−4t ((e11 + e44 − 2xe34 )∂t + (e34 − e21 − 2x(e11 + e33 ))∂x +
(2x2 − 4)e11 − 8xe12 + 3xe21 + (4x2 − 2)e22 − 2x2 e33 + xe43 ),
Σ5 = e2t ((e11 + e22 + e33 + e44 )(∂x + x) − 2e34 ),
Σ6 = e−2t ((e11 + e22 + e33 + e44 )(∂x − x) + 2e12 ),
Σ7 = (e11 + e44 )∂t + (e43 − e21 )∂x + x(e21 + e43 ),
Σ8 = (e22 + e33 )∂t + (e21 − e43 )∂x − x(e21 + e43 ),
Σ9 = e11 + e22 ,
Σ10 = e33 + e44 ,
Σ11 = e2t (e34 ∂t − (e22 + e44 )∂x − e21 + 2e34 − x(e22 + e44 )),
Σ12 = e−2t (e34 ∂t − (e22 + e44 )∂x − e21 + x(e44 + 2e33 − e22 )).
(60)
The 12 fixed block-antidiagonal symmetry operators can be conveniently presented as
Σ13 = e6t ((e13 − e41 − 4xe32 )∂t + (4x(e13 − e31 ) − (e23 + e41 ))∂x +
(8x2 + 4)e13 − (12x + 8x3 )e14 − 3xe23 + (4x2 + 2)e24 + 4x2 e31 + xe41 ),
Σ14 = e−6t ((e24 − e31 − 4xe32 )∂t + (e23 + e41 + 4x(e42 − e24 ))∂x +
−3xe23 + 4x2 e24 + 4(1 − x2 )e31 + (12x − 8x3 )e32 + xe41 + 2e42 ),
Σ15 = e4t (−e32 ∂t + (e13 − e31 )∂x + 3xe13 − (4x2 + 2)e14 − e23 + 2xe24 + xe31 ),
Σ16 = e−4t (e32 ∂t + (e24 − e42 )∂x + e23 − xe24 + 2xe31 + (4x2 − 2)e32 + xe42 ),
Σ17 = e2t ((e24 − e31 )∂t + e23 ∂x + x(e23 + e41 ) − 2e31 ) ,
Σ18 = e2t ((e13 − e42 − 2xe32 )∂t − (e23 + e41 + 2x(e31 − e13 ))∂x + x(e41 − e23 ) + 2x2 (e13 + e31 )),
Σ19 = e2t (e13 + e24 − 2xe14 ),
Σ20 = e−2t ((e13 − e42 )∂t − (e23 + e41 )∂x − 2e13 + x(e23 + e41 )),
Σ21 = e−2t (e24 − e31 − 2xe32 )∂t + (−2x(e24 − e42 ) + e23 + e41 )∂x + 2x2 (e24 + e42 ) + x(e41 − e23 ) ,
Σ22 = e−2t (e31 + e42 + 2xe32 ),
Σ23 = (e13 + e24 − e31 − e42 )∂x + (e13 − e24 + e31 − e42 )x,
Σ24 = e32 ∂t + (e24 − e42 )∂x + e23 − x(e24 + e42 ).
The 4 (two block-diagonal and two block-antidiagonal) extra sets of symmetry operators de-
(61)
14
CBPF-NF-004/16
pending on the unconstrained functions zi (x, t), i = 1, 2, 3, 4, are
e 1,z (x,t) = z1 (x, t)((e12 + e34 )∂t + (e11 − e22 + e33 − e44 )∂x − (e21 + e43 ) − x(e11 + e22 − e33 − e44 )),
Σ
1
e 2,z (x,t) = z2 (x, t)((e12 − e34 )∂x + e11 − e33 + x(e12 + e34 )),
Σ
2
e
Σ3,z (x,t) = z3 (x, t)((e14 − e32 )∂t + (e13 − e24 − e31 + e42 )∂x − e23 + e41 + x(e13 + e24 + e31 + e42 )),
3
e 4,z (x,t) = z4 (x, t)((e14 + e32 )∂x + e13 + e31 + x(e32 − e14 )).
Σ
4
(62)
e is recovered from Σ
e 3,z (x,t) by setting z3 (x, t) = 1. We have,
The Lévy-Leblond operator Ω
3
indeed,
e = Σ
e 3,z (x,t)=1 .
Ω
3
(63)
Some of the block-diagonal symmetry operators have special meaning: the identity I4 is given
by the combination I4 = Σ9 + Σ10 , while the fermion-number symmetry operator Nf is the
difference Nf = Σ9 − Σ10 . The time-derivative is a symmetry operator and, as it is the case for
the Schrödinger’s equation of the harmonic oscillator, it can be used to define a grading operator
D. We can set
1
1
(Σ7 + Σ8 ) = I4 · ∂t .
(64)
D =
4
4
A natural Schrödinger symmetry algebra is encountered. The identification (up to normalization) of its symmetry generators at degree ±1, ± 12 , 0 is given by
+1 : Σ1 + Σ2 ≡ H,
+ 12 : Σ5 ≡ P+ 1 ,
2
0 : D, C ≡ I4 ,
− 21 : Σ6 ≡ P− 1 ,
2
−1 : Σ3 + Σ4 ≡ K.
(65)
Σ5 ≡ P 1 , Σ6 ≡ P− 1 are, respectively, the creation/annihilation operators, while H, D, K are the
2
2
sl(2) subalgebra generators.
A first difference with respect to the symmetry algebra of the free Lévy-Leblond equation
is already encountered at this level. In the free case the Lévy-Leblond operator possesses halfinteger grading (= 21 ) with respect to the sl(2) Cartan generator of the Schrödinger subalgebra.
e has zero grading
In the present case Ω
e = 0.
[D, Ω]
(66)
This implies, as a a corollary, that for the quadratic potential there is no osp(1|2) symmetry
superalgebra induced by Ω± 1 .
2
The extra question to be investigated is whether, in the presence of a non-trivial potential,
the Lévy-Leblond operator admits a super-Schrödinger invariance.
In principle one can repeat the same steps as in the free case and compute the most gene = ΦΛ (x, t) · Ω.
e On the other hand,
eral solution obtained from the anti-commutators {Λ, Ω}
since these computations are rather cumbersome, it is preferable to address the question with
a different procedure, based on the key observation that the existence of the super-Schrödinger
symmetry requires the presence of at least one operator which is the square root (up to normalization) of H = Σ1 + Σ2 . Requiring Q2 = kH for some real k 6= 0 produces the most general
solution


0
∂t + 2x∂x + 2x2 r3
−2r3 x
 r1 r2

0
0
−r1 r3
 , (67)
Q = e2t 
2
 0
0
2r2 x r1 ∂t + 2r1 x∂x + (2r1 − 4r2 )x + 2r1 
0
0
r2
−2r2 x
15
CBPF-NF-004/16
depending on three real parameters ri (i = 1, 2, 3) and with the identification k = r1 r2 . Therefore, both r1 , r2 need to be non-vanishing. Once identified the most general operator Q, we need
to verify whether it belongs to the class of Lévy-Leblond symmetry operators. This means that
a matrix-valued function Φ(x, t) should be found for
either
e = Φ(x, t) · Ω
e or {Q, Ω}
e = Φ(x, t) · Ω.
e
[Q, Ω]
(68)
It is easily checked that, for r1,2 6= 0, no such matrix-valued function exists, implying that the
r1,2 6= 0 operators in (67) are not symmetry generators of the Lévy-Leblond operator (57) with
non-trivial potential.
In the presence of a non-trivial potential there is no analog of the osp(1|2) symmetry generated by the Q± 1 symmetry operators of the free case.
2
In the free case we discussed three separated osp(1|2) symmetry superalgebras, induced by
P± 1 , Q± 1 , Ω± 1 , respectively. In the presence of the non-trivial potential we obtain a single
2
2
2
osp(1|2) symmetry from the (anti)commutators of the creation/annihilation operators P± 1 en2
tering (65), with the even sector being given by P±1 = 2P±2 1 , P0 = {P+ 1 , P− 1 }. All these
2
2
2
e
operators commute with Ω:
e Ps ] = 0,
[Ω,
1
s = 0, ± , ±1.
2
(69)
Together with the extra generator H, D, K, C in (65), we therefore find a symmetry superalgebra
e given by u(1)⊕sl(2)⊃
for Ω,
+osp(1|2). It contains two odd generators P± 1 , while the 7 remaining
2
generators span the even sector.
The investigation and identification of other closed finite symmetry (super)algebras recovered
from the (60,61,62) symmetry operators is beyond the scope of the present paper.
7
Conclusions
We highlight the main results of the paper.
We proved, for the Lévy-Leblond square root of the free heat or the free Schrödinger equation,
the existence of a symmetry of first-order and second-order differential operators closing a finite
Z2 × Z2 -graded Lie superalgebra. Let’s take, for the sake of clarity, the 1 + 1-dimensional
Schrödinger equation. It can be equivalently written in two forms, either
(i∂t + ∂x2 )Ψ(x, t) = 0,
(70)
i∂t Ψ(x, t) = −∂x2 Ψ(x, t),
(71)
or
where 2 × 2 diagonal operators (the identity I2 is dropped for simplicity) act upon the 2-column
complex vector Ψ(x, t).
Three independent sets of osp(1|2) symmetry superalgebras are encountered. They are generated by the first-order, odd, symmetry differential operators Ω± 1 , P± 1 , Q± 1 , respectively.
2
2
2
Ω+ 1 is the Lévy-Leblond operator we initially started with; it is the square root of the
2
operator entering (70). P+ 1 (Q+ 1 ) is the square root of the operator entering the right hand
2
2
side (respectively, the left hand side) of equation (71). We investigated the mutual compatibility
conditions for these osp(1|2) superalgebras. Since the Ω± 1 operators commute with the P± 1
2
2
CBPF-NF-004/16
16
operators, a first Z2 × Z2 -graded Lie superalgebra, given by (44), is encountered. Its G11 sector
is empty, while its vector space is spanned by the Ps and Ωs0 symmetry operators (s, s0 =
± 21 , ±1, 0).
The Ω± 1 operators, together with the Q± 1 operators, produce an algebra containing higher2
2
order (degree ≥ 3) differential operators.
A finite, closed, non-trivial Z2 × Z2 -graded Lie superalgebra (spanned by the 13 generators
recovered from (53)) is discovered by requiring the presence of the two pairs of operators, P± 1
2
and Q± 1 .
2
The (70,71) system also possesses a (maximally N = 1) Z2 -graded super Schrödinger algebra
symmetry.
We completed the (1+1)-dimensional investigation by looking at the symmetries of the LévyLeblond square root of the (1 + 1)-dimensional heat equation with quadratic potential. This syse possesses a symmetry closing the Schrödinger altem, defined by the Lévy-Leblond operator Ω,
gebra sch(1). We proved, on the other hand, by exhaustive computations, that the non-vanishing
potential does not allow a graded extension (neither Z2 - nor Z2 ×Z2 -) of the Schrödinger algebra.
e commutes with the
The main difference, with respect to the free case, is due to the fact that Ω
e is a 0-grade
Cartan generator D of the sl(2) ⊂ sch(1) subalgebra, see (66). Measured by D, Ω
operator. In the free case, the Lévy-Leblond operator Ω+ 1 possesses the half-integer + 12 -grade
2
when measured by the corresponding D operator, see (36).
Some comments are in order. To our knowledge, this is the first time that a Z2 × Z2 graded Lie (super)algebra is encountered in the context of symmetries of partial differential
equations. This feature could prove to be significant for the community of physicists (searching
for applications of) and mathematicians (developing the mathematical structure of) graded color
Lie (super)algebras. It opens the possibility, e.g., to use them as spectrum-generating algebras
for suitable dynamical systems.
The constructions of (both Z2 - and Z2 × Z2 -) graded extensions of Lie symmetries require
two types of symmetry operators, the ones obtained from the commutator condition (14) and
those obtained from the anticommutator condition (15).
In Appendix B we constructed a Z2 × Z2 -graded Lie superalgebra symmetry of the LévyLeblond operator associated with the free heat equation in (1 + 2)-dimensions. A new feature
appears. The Z2 × Z2 -graded symmetry allows to explain the existence of first-order differential
symmetry operators which do not belong to the two-dimensional super Schrödinger algebra.
It is easily realized that our results have a more general validity, so that (70,71) and (B.1)
are the prototypes of a class of Z2 × Z2 -graded invariant equations. The formal demonstration,
based on the systematic Clifford algebra tools discussed in section 2, of the existence of a
Z2 × Z2 -graded Lie superalgebra symmetry for the Lévy-Leblond square roots of the free heat
or Schrödinger equations in (1 + d)-dimension, for an arbitrary number of space dimensions d,
will be presented elsewhere. We are reporting the Z2 × Z2 -graded symmetry of the original
Lévy-Leblond equation (the square root of the (1 + 3)-dimensional free Schrödinger equation)
in [22], a paper based on a talk given at the 2016 ICGTMP (Group 31 Colloquium).
Finally, as discussed in Appendix C, future investigations of Z2 × Z2 -graded symmetries
of partial differential equations should systematically study the different possible graded Lie
(super)algebras recovered from different assignment of gradings to the symmetry operators.
17
CBPF-NF-004/16
Acknowledgments
Z.K. and F.T. are grateful to the Osaka Prefecture University, where this work was initiated, for hospitality. F.T. received support from CNPq (PQ Grant No. 306333/2013-9). N. A.
is supported by the grants-in-aid from JSPS (Contract No. 26400209).
Appendix A: On Z2 × Z2 -graded color Lie (super)algebras and
their relations with quaternions and split-quaternions
To make the paper self-contained we collect here the main properties of the Z2 × Z2 -graded
color Lie algebras and superalgebras induced by a composition law; we make also explicit their
relations with both the division algebra of quaternions and its split version.
A Z2 × Z2 -graded color(super)algebra G is decomposed according to
G = G00 ⊕ G01 ⊕ G10 ⊕ G11 .
(A.1)
For the generators xα~ , xβ~ ∈ G, with α
~ , β~ labeling the grading, the G × G → G brackets are
~
(xα~ , xβ~ ) = xα~ xβ~ − (−1)(~α·β) xβ~ xα~ ,
(A.2)
where xα~ xβ~ is a composition law and, for the Z2 × Z2 -graded color algebra, the inner product
is defined by
~ = α1 β2 − α2 β1 ,
(~
α · β)
(A.3)
while, for the Z2 × Z2 -graded color superalgebra, the inner product is given by
~ = α1 β1 + α2 β2 .
(~
α · β)
(A.4)
For both Z2 × Z2 -graded color algebra and superalgebra, the grading deg(xα~ , xβ~ ) of the (xα~ , xβ~ )
bracket is
~
deg(xα~ , xβ~ ) = α
~ + β.
(A.5)
Z2 × Z2 -graded Jacobi identities are imposed, in both algebra and superalgebra cases, through
~
~
(−1)(~α·~γ ) (xα~ , (xβ~ , x~γ )) + (−1)(β·~α) (xβ~ , (x~γ , xα~ )) + (−1)(~γ ·β) (x~γ , (xα~ , xβ~ )) = 0.
(A.6)
Some comments are in order. In both algebra and superalgebra cases, the G00 -graded sector of
G is singled out with respect to the three other sectors. For what concerns the three remaining
sectors we have that
- for the color algebra, all three sectors G01 , G10 and G11 are on equal footing while,
- for the color superalgebra, the G11 sector is singled out with respect to the G01 and G10 sectors
(which are equivalent and can be interchanged).
The basic examples of Z2 × Z2 -graded Lie algebras and superalgebras are induced by the
quaternions and the split-quaternions. We recall here their basic features.
The division algebra of the quaternions (over R) is given by the generators e0 (the identity)
and the three imaginary roots ei , i = 1, 2, 3, with composition law
ei · ej
= −δij e0 + ijk ek ,
(A.7)
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for the totally antisymmetric tensor 123 = 231 = 312 = 1.
One should note that the three imaginary quaternions are on equal footing.
The split-quaternions (over R) are given, see [23], by the generators ee0 (the identity) and eei ,
i = 1, 2, 3, with composition law
eei · eej
= −ηij ee0 + e
ijk eek .
(A.8)
The metric ηij is diagonal and satisfies
η11 = η22 = −η33 = 1,
(A.9)
while the totally antisymmetric tensor e
ijk is
e
123 = −e
231 = −e
312 = −1.
(A.10)
For split-quaternions, only the generators ee1 , ee2 are on equal footing and can be interchanged.
Different Lie (super)algebras or color Lie (super)algebras can be defined in terms of both
quaternions and split-quaternions; their brackets are given by either commutators or anticommutators obtained from the composition laws ((A.7) for quaternions and (A.8) for splitquaternions). In all cases the graded Jacobi identities are satisfied.
Starting from either the quaternions or the split-quaternions, the following algebraic structures can be defined:
i) a Lie-algebra structure with brackets defined by the commutators;
ii) Z2 -graded Lie algebra structures with brackets defined by the appropriate (anti)commutators;
iii) a Z2 ×Z2 -graded color Lie algebra structure defined by the (anti)commutators given by (A.2)
and (A.3);
iv) Z2 ×Z2 -graded color Lie superalgebras structures defined by the appropriate (anti)commutators
given by (A.2) and (A.4) .
In all cases the identity (either e0 or ee0 ) belongs to the even sector (G00 for the Z2 ×Z2 gradings).
An important remark is that in the quaternionic case, since all three imaginary roots are on
equal footing, there exists a unique assignment of the graded (super)algebras in all four cases
above. Without loss of generality we can assign, for the ii case, e0 , e3 ∈ G0 and e1 , e2 ∈ G1 ;
for both iii and iv cases the assignment, without loss of generality, can be assumed to be
e1 ∈ G01 , e2 ∈ G10 , e3 ∈ G11 .
For split-quaternions, since ee3 is singled out with respect to ee1 , ee2 , a unique assignment is
only present in the Lie algebra cases i and iii. In the supercases ii and iv the assignment of the
grading depends on a θ-angle. We have indeed that
- for the Z2 superalgebra case, inequivalent consistent assignments are given by ee0 , cos θe
e3 +
sin θe
e1 ∈ G0 and ee3 , − sin θe
e3 + cos θe
e1 ∈ G1 ;
- for the Z2 × Z2 color superalgebra case, inequivalent consistent assignments are given by
cos θe
e3 + sin θe
e1 ∈ G11 , − sin θe
e3 + cos θe
e1 ∈ G01 , ee2 ∈ G10 .
Appendix B: Z2 × Z2 -graded Lie symmetry of the free (1 + 2)dimensional Lévy-Leblond equation
We present, for completeness, the construction of the Z2 × Z2 -graded Lie superalgebra symmetry of the Lévy-Leblond equation associated with the free (1 + 2)-dimensional heat equation.
For this case the Lévy-Leblond operator is realized by 4 × 4 matrices. Five 4 × 4 gammamatrices γµ (µ = 1, 2, . . . , 5), satisfying {γµ , γν } = 2ηµν I, with ηµν a diagonal matrix with
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CBPF-NF-004/16
diagonal entries (+1, +1, +1, −1, −1), can be introduced. We have γ12 = γ22 = γ32 = −γ42 =
−γ52 = I. We also set γ± = 21 (γ3 ± γ4 ). Expressed in the notations introduced in Section 2, an
explicit representation is given, e.g., by γ1 = Y X, γ2 = Y X, γ3 = XI, γ4 = AI, γ5 = Y A.
The (1 + 2)-dimensional free Lévy-Leblond equation, defined by the operator Ω, is given by
ΩΨ(x1 , x2 , t) = 0,
Ω = γ+ ∂t + λγ− + γi ∂i .
(B.1)
In the above formula i = 1, 2, the Einstein convention over repeated indices is understood
and ∂i = ∂xi . In the following we also make use of the antisymmetric tensor ij , with the
normalization convention 12 = −21 = 1.
The free heat equation reads
Ω2 Ψ(x1 , x2 , t) = (λ∂t + ∂i2 )Ψ(x1 , x2 , t) = 0.
(B.2)
The following list of first-order differential symmetry operators, satisfying the (14) commutation
property with Ω, is easily derived
H = I∂t ,
1
1
1
D = −I(t∂t + xi ∂i + ) − γ+ γ− ,
2
2
2
λ
1
K = −I(t2 ∂t + txi ∂i − x2i + t) − xi γ+ γi − tγ+ γ− ,
4
2
P+i = I∂i ,
1
1
P−i = I(t∂i − λxi ) + γ+ γi ,
2
2
1
J = ij (Ixi ∂j + [γi , γj ]),
8
e = ij (γ+ γi ∂j + λ [γi , γj ]),
X
8
C = I.
(B.3)
D, K are the only operators of the above list which do not commute with Ω ([D, Ω] = 12 Ω,
[K, Ω] = tΩ).
D is the scaling operator. For a given operator Z its scaling dimension z is defined via
the commutator [D, Z] = zZ. The scaling dimensions of the above operators are given by
e = z(C) = 0.
z(H) = +1, z(K) = −1, z(P±i ) = ± 12 , z(D) = z(J) = z(X)
The operators D, H, K close a sl(2) symmetry algebra; the operators P±i , C close the 2dimensional Heisenberg-Lie algebra h(2). J is the generator of the SO(2) rotational symmetry.
These 3 + 5 + 1 = 9 operators span the 2-dimensional Schrödinger symmetry algebra sch(2).
e entering (B.3) is better grasped
The physical interpretation of the remaining operator X
when introducing the Z2 × Z2 -graded extension. For that we need to introduce symmetry
operators satisfying the (15) anticommutation property with Ω. We get
Q+ = γ+ ∂t − λγ− ,
1
Q− = γ+ (t∂t + xi ∂i + 1) − λxi γi − λtγ− ,
2
1
Xi = γ+ ∂i − λγi .
2
(B.4)
The operators Q+ , Xi anticommute with Ω ({Q+ , Ω} = {Xi , Ω} = 0), while {Q− , Ω} = −γ+ Ω.
Their scaling dimension (measured by D) are z(Q± ) = ± 21 , z(Xi )=0.
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The operators Q± belong to the odd sector of a osp(1|2) symmetry superalgebra whose even
sector is the sl(2) algebra spanned by H, D, K. We get, indeed,
{Q+ , Q+ } = −2λH,
{Q+ , Q− } = 2λD,
{Q− , Q− } = 2λK.
(B.5)
The closure of the full set of osp(1|2) anticommutation relations follows immediately.
We obtain, as symmetry Lie superalgebra for Ω, the N = 1 supersymmetric extension ssch(2)
of the two-dimensional Schrödinger algebra by adding, to the even generators spanning sch(2),
an odd sector containing the operators Q± . This requires, for consistency, to add the operators
Xi to the odd sector, since they are recovered from the commutators
[P±i , Q∓ ] = ±Xi .
(B.6)
The N = 1 Schrödinger symmetry superalgebra ssch(2) is decomposed according to
ssch(2) = ssch(2)0 ⊕ ssch(2)1 ,
ssch(2)0 = {H, D, K, P±i , C, J},
ssch(2)1 = {Q± , Xi }.
(B.7)
It is a straightforward exercise to check the closure of the Z2 -graded (anti)commutation relations
derived for the operators entering ssch(2).
e entering (B.3) is still unaccounted for. The operator
One should note that the operator X
e on the other hand, naturally appears in the Z2 × Z2 -graded extension, being obtained from
X,
the Xi ’s commutators; we have indeed
e
[Xi , Xj ] = λij X.
(B.8)
We recall that, in the Z2 -graded ssch(2) algebra, the Xi ’s brackets are given by the anticommutators
{Xi , Xj } =
1 2
λ δij C.
2
(B.9)
We therefore see that the introduction of the Z2 × Z2 -graded structure offers a proper interpretation to all first-order differential operators entering (B.3) and (B.4).
The construction of the Z2 × Z2 -graded Lie superalgebra goes as follows. It requires the
introduction of extra first-order and second-order differential operators Ps,ij (with s = ±1, 0)
and X ±ij , introduced via the anticommutators
P1,ij = {P+i , P+j },
P0,ij = {P+i , P−j },
P−1,ij = {P−i , P−j }
(B.10)
and
X ±,ij
= {P±i , Xj }.
(B.11)
Their scaling dimensions are, respectively, z(Ps,ij ) = s and z(X ±,ij ) = ± 21 .
We observe that the anticommutators {Q± , Xi } = −λP±i produce the first-order differential
operators P±i . New first-order differential operators, not entering (B.3) and (B.4), are also
obtained. We have, for instance,
e = {P+2 , X1 } − {P+1 , X2 } = X +,21 − X +,12 = λ(γ2 ∂1 − γ1 ∂2 ).
[Q+ , X]
(B.12)
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The Z2 × Z2 -graded symmetry structure is introduced as a Z2 × Z2 -graded Lie superalgebra,
with brackets, the (anti)commutators, defined in (A.2) and based on the (A.4) inner product.
The vector space of the Z2 × Z2 -graded Lie superalgebra LZ2 ×Z2 is decomposed according to
LZ2 ×Z2
= L00 ⊕ L01 ⊕ L10 ⊕ L11 ,
(B.13)
with the different sectors respectively spanned by the operators
e Ps,ij },
L00 = {H, D, K, J, X,
L01 = {P±i },
L10 = {Q± , X ±,ij },
L11 = {Xi }.
(B.14)
With lenghty and tedious, but straighforward, computations the following three main properties
are proven:
i) all operators in (B.14), including the second-order differential operators, are symmetry operators, satisfying either the (14) or the (15) conditions with respect to the operator Ω given in
(B.1),
ii) the (A.2,A.4) brackets, defined by the respective (anti)commutators, are closed on the (B.14)
generators and, finally,
iii) the Z2 × Z2 -graded Jacobi identities (A.6) are satisfied by the (B.14) generators.
These three main results prove that LZ2 ×Z2 is a Z2 × Z2 -graded Lie superalgebra symmetry
of the Lévy-Leblond square root of the (1 + 2)-dimensional free heat equation.
By adding the identity operator C to the L00 sector of LZ2 ×Z2 we obtain the Z2 × Z2 -graded
Lie superalgebra L0Z2 ×Z2 , given by
L0Z2 ×Z2
= LZ2 ×Z2 ⊕ u(1).
(B.15)
Appendix C: On different grading assignments for the symmetry operators
In Appendix A we showed, with examples taken from quaternions and split-quaternions, that
different Z2 × Z2 -graded Lie (super)algebras can be defined on the same vector space, based on
a different grading assignment for the same given set of operators.
It is tempting to investigate this feature for the symmetry operators of the free Lévy-Leblond
equation. Since, in particular, the natural finite Z2 × Z2 -graded Lie superalgebra (53) is induced
from the repeated (anti)commutators of the operators P± 1 ∈ G01 , Q± 1 ∈ G10 , one can ask
2
2
which is the output if P± 1 (given in (26,31)) are kept in the G01 sector, while Q± 1 (given in
2
2
(45,46)) are assigned to the G11 sector of a Z2 × Z2 -graded Lie superalgebra. The consistency
condition requires to derive the operator X ∈ G10 from the commutators [±P± , Q± ] = X. The
commutator [Q+ , Q− ] = Q0 produces the first-order differential operator Q0 ∈ G00 . The further
commutators [P± 1 , Q0 ] produce the extra (41) pair of Ω± 1 operators. One should note that Ω± 1
2
2
2
are assigned to G01 and that Ω+ 1 is the original Lévy-Leblond operator introduced in equation
2
(22).
The construction has to be continued. The new grading assignment produces a Z2 ×Z2 -graded
Lie superalgebra which, unlike (53), is infinite-dimensional and spanned by differential operators
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of any order. To check this statement it is sufficient to compute the repeated commutators
[Q+ 1 , [Q+ 1 , . . . [Q+ 1 , Q0 ] . . .]].
2
2
2
For completeness we produce here a finite Z2 × Z2 -graded Lie superalgebra, defined abstractly in terms of (anti)commutators and whose consistency is implied by the fact that the
graded Jacobi identities are satisfied. This algebra cannot be realized by the Lévy-Leblond symmetry operators. It shares, nevertheless, some features in common. We impose, in particular,
the existence of two osp(1|2) subalgebras, whose generators (as in Section 5) are expressed as
{P0 , P± 1 , P±1 } and {H, D, K, Q± 1 }. Their respective (anti)commutators are given by equations
2
2
(29,32,47,48,49). Equation (50) introduces the extra generator X. An extra set of generators,
R0 , R±1 , Y, Z, C, are introduced through the positions
R0 = [Q 1 , Q− 1 ],
2
2
R±1 = {Q± 1 , P± 1 },
2
2
Y = {Q− 1 , P 1 },
2
2
Z = {Q 1 , P− 1 },
2
(C.1)
2
while C is the center of the algebra.
A consistent Z2 × Z2 -graded Lie superalgebra assignment is realized by spanning the graded
vector spaces as follows:
G00 = {H, D, K, P±1 , P0 , R0 , C},
G01 = {P± 1 },
2
G10 = {X, R±1 , Y, Z},
G11 = {Q± 1 }.
(C.2)
2
The remaining non-vanishing (anti)commutators which define the Z2 × Z2 -graded Lie superalgebra structure of (C.2) are given by
[H, Y ] = R1 ,
[H, Z] = R1
[H, R−1 ] = Y + Z,
[D, R±1 ] = ±R±1 ,
[K, R1 ] = Y + Z,
[K, Y ] = R−1 ,
[K, Z] = R−1 ,
[R0 , X] = Y − Z,
{X, X} = C,
{X, R±1 } = −P±1 ,
{X, Y } = −P0 ,
{X, Z} = −P0 ,
(C.3)
{X, Q± 1 } = −P± 1 .
2
2
These (anti)commutators guarantee the closure of the graded Jacobi identities for (C.2).
Unlike (53), the finite Z2 × Z2 -graded Lie superalgebra (C.2) is not a symmetry algebra of
the Lévy-Leblond equation. It is an open question whether it appears as the symmetry algebra
of some dynamical system.
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